What does the Bible say about artificial hair or style?

Thanks for your question.  The first thing I would say is that the Bible was never meant to give a specific answer to every kind of question (such as the wearing of artificial hair).  The Old Testament Law of Moses had ten basic commandments and then the application of these commandments was spelled out in greater detail in Deuteronomy and Exodus.  Even these ten commandments were summarized in two main points:  (1) love the Lord with all your heart, soul, mind, and strength; and, (2) love your neighbor as yourself.  Thus, the over arching guiding principle for our relationship with men is that of love.  And so it is that Paul can write,

8 Owe no one anything, except to love one another, for the one who loves his neighbor has fulfilled the law. 9 For the commandments, “Do not commit adultery, do not murder, do not steal, do not covet,” (and if there is any other commandment) are summed up in this, “Love your neighbor as yourself.” 10 Love does no wrong to a neighbor. Therefore love is the fulfillment of the law (Romans 13:8-10, NET  Bible).

The Jews of Jesus’ day sought to create a rule for every particular situation, and thus there were (and still are) literally hundreds of rules.  That is simply not the way God intended it to be.

So, having said this, not only is there no text that specifically answers your question, there is no need for such a rule.  The reason is that God’s Word gives us the few commands and prohibitions that are necessary, and then gives us guiding principles by which to deal with other issues.  So, what might some of these guiding principles be?  I’m sure there are a number of them, but allow me to mention a few.

(1)    Whatever we do (or don’t do) should be to the glory of God (1 Corinthians 10:31).

(2)    Whatever we do should promote/advance the gospel (1 Corinthians 9:19-23).

(3)    Whatever we do should edify (build up) others, and not cause them to stumble (1 Corinthians 8:7-13; 10:23-24).

(4)    With regard to our hair, Paul taught that it was given for a covering.  He also says that it is shameful for a woman to cut off all her hair (1 Corinthians 11:1-6).  Thus, if a woman had lost her hair because of chemotherapy (or some other malady not within her control) I would think that she would want to purchase a wig, at least until her hair grew in.

(5)    Paul and Peter warn women against seeking the kind of adornment which seeks to gain the attention of others, rather than adorning their husband with the headship God says should be his (1 Corinthians 11:1-16; 1 Peter 3:1-6).

(6)    Matters that are not specifically addressed in the Bible, and for such matters there is no clear governing principle, which either commends or condemns it.   (For example, should a Christian bowl, or play cards?)  Such matters would be issues of personal conviction.  One may do this, or not do so, based upon conscience.  We ought not debate such practices, nor should we condemn others or seek to convert them to our personal convictions (Romans 14:1—15:7).

Personally, I see the issues you have asked about as matters of personal conviction, and thus I am not surprised that the Bible neither condemns nor commands in these matters.  The fact that the Bible does not address these questions directly (by command or prohibition) or indirectly (by inference or by principle) is instructive to me.